Polynomially convex hulls with piecewise smooth boundaries

Abstract

is the boundary of a compact convex set Y(z) c C m with nonempty interior. Set Y= U {z} x Y(z). Alexander and Wermer [1] and Slodkowski [8] proved zcbD independently that each point p in the polynomial convex hull ~ of M, n (p) ~D, lies in the graph of a bounded analytic disk F(z) = ( z , f 1 (z) , . . . ,f~ (z)) ( f j~ H | (D) for l < j < m) with image in 3~r. Alexander and Wermer obtained a more precise information in the case when m = 1 and each fiber M(z) is a circle [1, Theorems 2 and 3]. In this paper we shall assume the existence of an analytic disk F o (z) = (z, f0 (z)) on D, continuous on / ) , such thatfo (z) is an interior point of Y(z) for each z ~ bD. Then each bounded analytic disk F ( z ) = (z , f (z)) in AI which passes through a boundary point of A~r lying over D is entirely contained in the boundary o fAir and its cluster set F(D) \ F ( D ) is contained in M (Theorem 1.1). If m = 1 and M is a smooth submanifold of r as above, then the hull 3~r has piecewise smooth boundary and is f'dled by the images of analytic disks with smooth boundary values contained in M (Theorem 1.2). We also obtain a uniqueness result for the representing measures (Corollary 1.3).